
The special case of \(q=N^{p}\)  with \(p=1/2\)  is called the double-scaling limit of SYK and was studied more fully in [1]}[2]}. I thank Henry Lin for pointing this out.

I can't recall who it was, but after this paper was circulated someone suggested a similarity with mechanisms discussed by Silverstein and collaborators [3]}[4]} in their discussion of the static patch. Consider going from
low temperature at fixed \(q\)  (the conformal limit) to the limit \(T\rightarrow \infty , \ q\rightarrow N^p\)  in two steps. First, keeping \(q\)  fixed we increase \(T\)  which has the effect of pulling in the boundaries as in figure REF , but without driving the system into the hyperfast regime. This pushes the theory from a low temperature near-extremal black hole to a higher temperature non-extremal black hole, but not toward de Sitter.
Then we let \(q\)  increase. That is what brings the system from black hole behavior to a de Sitter-like phase.
Compare this with Silverstein and collaborators who first deform a holographic CFT by a \(\bar{T} T\)  deformation. If I understand correctly, this has the effect of pulling in the boundaries in much the same way as a cutoff; again illustrated by figure REF . In a certain sense it is also increases the temperature, or more exactly the ratio of the temperature to the cutoff scale of the CFT. Of course at the end there is not much left of the CFT.
If the \(\bar{T} T\)  deformation is anything like a conventional cutoff this makes the theory spatially non-local but does not affect the k-locality of the couplings. (The couplings in an ordinary gauge theory are at most quartic which means the theory is 4-local.)
They then follow this operation by what they call a \(\Lambda \)  deformation, which, it is argued, drives the system to a de Sitter phase. If it does produce de Sitter space with a inflating interior, then by the arguments in this paper it should be possible to show that the \(\Lambda \)  deformation leads to a breakdown of k-locality, possibly by the introduction increasingly high powers of the fields, and to hyperfast scrambling.

Herman Verlinde has called to my attention a number of talks in which he also conjectured a relation between SYK in the double scaling region and de Sitter space [5]}. See also Akash Goel and Herman Verlinde, in preparation.

